Second Order Symmetric and Maxmin Symmetric Duality with Cone Constraints

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1 International Journal of Oerations Research International Journal of Oerations Research Vol. 4, No. 4, ) Second Order Smmetric Mamin Smmetric Dualit with Cone Constraints I. Husain,, Abha Goel, M. Masoodi 3 Deartment of Mathematics, Jaee Institute of Engineering echnolog, Guna, MP, India Deartment of Mathematics, G.G.S. Indrarastha Universit, Delhi, India 3 Deartment of Statistics, Kashmir Universit, Srinagar, Kashmir, India Received October 5; Revised August 6; Acceted March 7 AbstractIn this aer, a air of second order smmetric dual rograms with cone constraints is formulated. For this air of rograms, weak, strong, converse self dualit theorems are validated under bonveit-boncavit condition. Further, a air of second order mamin mied integer smmetric dual rograms involving cones are constructed for this air of rograms, smmetric as well as self dualit is investigate. Some articular cases are derived from our results. KewordsSecond order smmetric self dualit, Cone constraints, Dualit theorems, Bonveit boncavit, Mamin smmetric dualit, Mied integer smmetric dual rograms. INRODUCION Smmetric dualit in nonlinear rogramming in which the dual of the dual in the rimal was first studied b Dorn 96). Motivated with the results of Dorn 96), the notion of smmetric dualit was develoed significantl b Dantzig et al. 965), Chra Husain 98), Mond Weir 989). Dantzig et al. 965) formulated a air of Wolfe te smmetric dual rograms with the non-negative orthants as cones under conveit-convavit on the kernel function that occurs in the rograms. he same results were subsequentl generalized b Bazaraa Goode 973) to arbitrar cones. Na 988) studied smmetric dualit for a air of nonlinear mied integer rograms involving arbitrar cone under inveit. Mond 974) initiated second order smmetric dualit of Wolfe te in nonlinear rogramming also indicated ossible comutational advantages of second order dual over the first order dual. Later, Bector Chra 986) resented a air of Mond-Weir te second order dual rograms roved weak, strong self dualit theorems under seudobonveit seudoboncarit. Devi 998) constructed a air of second order smmetric dual rograms over cones studied dualit for the same; but this formulation of second order smmetric dual rograms seems quite strange aarentl different from the traditional Wolfe te second order smmetric dual rograms of Mond 974) as well as Mond-Weir te second order smmetric dual rograms formulated b Bector Chra 986). In 969) Balas resented a air of Wolfe te first order minima mied integer smmetric dual rograms as a generalization of the results of Dantzig et al. 965), while in 995) Kumar et al., Husian Chra 98) dealt with Mond-Weir te first order maimin mied integer smmetric dual rograms. Later, Gulati Ahmed 997) formulated second order maimin mied integer smmetric dual rograms roved various dualit theorems including self dualit theorem. In this research, we formulate Wolfe te second order dual rograms with cone constraints rove weak, strong, converse self dualit theorems under bonveit boncavit condition. Further, we generalize these Wolfe te dual rograms to maimin second order dual rograms b constraining some of the comonents of the two variables of the rograms to belong to arbitrar sets. For integers of these rograms also, smmetric as well as self dualit is incororated. Particular cases are generated from our results.. NOAIONS AND PRE-REQUISIES Let R k denote the k-dimensional Euclidian sace. Let Γ be a closed conve cone with nonemt interior in R k. Definition. he ositive olar cone Γ of Γ is defined b ξ R ξ, for all Γ = Γ where denotes the transose of. Let f, ) be a twice differentiable real valued function on an oen set in R n R m. hen f, ) Corresonding author s ihusain@ahoo.com 83-73X Coright 7 ORSW

2 Husain, Goel, Masoodi: Second Order Smmetric Mamin Smmetric Dualit with Cone Constraints IJOR Vol. 4, No. 4, ) f denote gradient vectors with resect to resectivel evaluated at, ). f, ) f are resectivel the n n m m smmetric Hessian matrices. f) i is the m m matri obtained b differentiating the elements of f with resect to i f ) denotes the matri whose i, j) the element is f ). j i Definition. Let C C be closed conve cones in R n R m resectivel. A twice differentiable function f : C C R is said to be i) Bonve in, if for all, q, u C fied f v, ) f uv, ) u, ) f uv, ) + f uvq, ) q f uvq, ) ii) Boncave in, if for fied for all,, v C f v, ) f v ) f + f, ) f, ) iii) Skew-smmetric, when both C C are in R n C = C = C sa), f = f, for all C C. In the sequel, we shall require the Fritz John te necessar otimalit conditions derived b Bazaraa Goode 973) which are embodied in the following roosition. Proosition. Let X be a conve set with nonemt interior in R n C be a closed conve cone in R m. Let F be real valued function G be a vector valued function, both defined on X. Consider the roblem: P ): Minimize Fz) Gz) C z X If z solves the roblem P ), then there eist α R δ C such that δ α Fz ) + Gz ) z z ) for all z X, δ Gz ) = α, δ) α, δ) he following concet of searabilit Balas 969)) is also needed in the subsequent analsis of this research. Definition 3. Let s, s,..., s be elements of an elementar vector sace. A real valued function,,..., H ) s s s will be called searable with resect to s if there eist real-valued function H s )indeendent of s,..., s ),..., H ) s s indeendent of s ), such that H s, s,..., s ) = H s ) + H s,..., s ). 3. SECOND ORDER SYMMERIC AND SELF DUALIY In this section, we consider a air of second order smmetric dual nonlinear rograms with cone constraints establish aroriate dualit theorems. Consider the following two rograms: Primal Problem SP): Minimize G,, ) = f, ) Dual Problem ) + f f f f f C ) C C ) SD): Maimize H, q, ) = f where ) + f f q q f, q ) f + f, q ) C 3) C C 4) i) f: C C R is a twice differentiable function, ii) C C are closed conve cones with nonemt interior in R n R m, resectivel, iii) C C are ositive olar cones of C C resectivel. heorem Weak dualit). Let,, ) u, v, q) be feasible solutions of SP) SD) resectivel. Assume that f, ) is bonve with resect to for fied f, ) is boncave with resect to for fied for all feasible,,, u, v, q).

3 Husain, Goel, Masoodi: Second Order Smmetric Mamin Smmetric Dualit with Cone Constraints IJOR Vol. 4, No. 4, ) hen infimum SP) suremum SD). Proof. B bonveit of f, ), we have f v, ) f uv, ) u) f uv, ) + f uvq, ) q f uvq, ) 5) b boncavit of f, ), we have f v, ) f v ) f + f f 6) Multiling 6) b ) adding the resulting inequalit to 5), we obtain f v, ) f + f, ) ) f f uv, ) u f uv, ) + f uvq, ) ) q f uvq, ) f uv, ) + f uvq, ) v f + f. 7) Now since C f uv, ) + f uvq, ) C, we have f uv f uvq + 8) since v C [ f + f] C, we have v f + f 9) he inequalit 7) together with 8) 9), ields, f [ f + f ] f f uv, ) u [ f uv, ) + f uvq, ) ] q f uvq, ) his imlies infimum SP) suremum SD). heorem Strong dualit). Let,, ) be an otimal solution of SP). Also let A ): the matri is non singular, f A ): f, ) ) be negative definite. hen,, q = ) is feasible for SD) the objective values of the rograms SP) SD) are equal. Moreover, if the requirements of heorem are fulfilled, then,, q ) is an otimal solution of SD). Proof. We use Proosition to rove this theorem. Here z =,, ), z =,, ), C, R m C ) Fz ) = f f + f, ) f, ) Gz ) = f + f, ) C = C Since,, ) is an otimal solution of SP), b Proosition, there eist α R β C such that α f α + β) f α α + + β) f, ) ) α + α + β) f α + α + + β) f, ) ) ) α + α + β) f = ) β f + f = ) α, β) 3) α, β) 4) he relation ), in view of the hothesis A ), gives β = α + ). 5) It follows that α, for if α =, 5) imlies β =. Hence α, β) = contradicts 4). hus α >. Now utting = using 5) in ), we obtain, α ) f, ) ) ), for all C. Putting = + using α >, from the above inequalit f ) which, because of A ), ields, = 6) Using 5) 6) along with α > in ) we have ), f for all C 7)

4 Husain, Goel, Masoodi: Second Order Smmetric Mamin Smmetric Dualit with Cone Constraints IJOR Vol. 4, No. 4, ) Since C is closed conve cone, therefore, for each C C, it imlies + C. Now, relacing b + in 7), we have ) f + f 8) his imlies f + f C. hus,, q = ) is feasible for SD). Putting = in 7) = in 8), we have resectivel ) f + ) f +. hese together imlies ) f + =. 9) Using have β = α = along with α > in ), we ) f + = ) Consequentl, we obviousl have, G,, ) = f f + ) f = f f +, q ) ) q f, q ) = H, q, ) hat is, the objective values of SP) SD) are equal. B heorem, the otimalit of,, z ) for SD) follows. We will onl state a converse dualit theorem heorem 3) as the roof of this theorem would follow analogousl to that of heorem. heorem 3 Converse dualit). Let,, q ) be an otimal solution of SD). Also let C ): the matri f, ) is nonsingular, C ): f, q ) ) be a ositive definite. hen,, = ) is feasible for SP) the objective values of SP) SD) are equal. Furthermore, if the hotheses of heorem are met, then,, ) is an otimal solution of SP). heorem 4 Self dualit). Let f: R n R m R be skew smmetric C = C, then SP) is self dual. Furthermore, if SP) SD) are dual rograms, s, ) is an otimal solution for SP), then,, = ) q=,, ) are otimal solutions for SP) SD), G,, ) = = H, q, ). Proof. Recasting the roblem SD) as a minimization roblem, we have SD) : ) Minimize f f + f, q ) q f, q ) f + f, q ) C C C. Since f is skew smmetric, f = f, ) f = f, ); C = C, the roblem SD) becomes Minimize f f + f, q )) q f, q ) f, ) f q, ) C C C which is just the rimal roblem SP). hus SP) is self dual. Hence if, q, ) is an otimal solution for SP), then conversel, Also, G,, ) = H, q, ). Now we shall show that G,, ) =. ) G,, ) = f f + f, ) f, ) ) Since C therefore, we have f f C,

5 Husain, Goel, Masoodi: Second Order Smmetric Mamin Smmetric Dualit with Cone Constraints IJOR Vol. 4, No. 4, ) ) f + f. ) Using ) in ), we have G,, ) f f. Using the conclusion = of heorem, we get MSD): Min Maψ, r, ),, r ) = f f + f, r ) ) f r ) f, r ) f + f, r ) K V,, ) S K 3 G,, ) f, ). 3) Similarl, in view of C together with f + f, q ) C, q =, we have H, q, ) f, ). 4) B heorem, we have f G,, ) = H, q, ) f, ). his imlies G,, ) = H q,, ) = f = f, ) = f, ). Consequentl, we have G,, ) =. 4. MAXMIN SYMMERIC AND SELF DUALIY n Let U V be two arbitrar sets of integers in R m R resectivel. Let K K be closed conve n n cones with nonemt interiors in m m R, R, resectivel. Let f, ) be a real valued function defined on n m an oen set in R R containing S where S = U K = V K. Let K i =, ) be the olars of K i. We consider the following air of nonlinear mied integer rograms: Primal Problem MSP): Ma Min φ, s, ), s, i ) = + f ) f f, s ) s f, s ) f f, s ) K U,, ) K. m m where s R r R n n. Also their feasible solutions will be denoted b A = s U K,, ),, f + f, r ) K B = r V S K,,,, f f, s ) K. heorem 5 Smmetric dualit). Let, s, ) otimal solution of MSP). Also, Let be an i) f, ) be searable with resect to or, ii) f, ) be bonve in for ever, ), boncave in for ever, ), iii) f, ) be thrice differentiable in, iv) f is non singular, v) f, s ) ) is negative definite. hen a) s = b) ) f = c) φ, s, = ) = ψ, r, = ), d), r, ) is an otimal solution of MSD) Proof. Let Z = Ma Min φ, s, ):, s, ) A, s, W = Min Ma ψ, r, ):, r, ) B,, r Since f, ) is searable with resect to or sa, with resect to ), it follows that f = f ) + f, ). 5) Dual Problem herefore, f = f, ) f = f, ).

6 Husain, Goel, Masoodi: Second Order Smmetric Mamin Smmetric Dualit with Cone Constraints IJOR Vol. 4, No. 4, ) 4 Now Z can be rewritten as, s, Z = Ma Min f ) + f, ) ) f, ) ) + f, s ) s f, s ) subject to f, ) f, s ) K or, ) KK, U V Ma Min Min f f = ) +,, s ) f, ) + f, s ) s f, s ), Z Ma Min f U V = ) +Θ ), 6) where Θ MSP) : ) ) Min f f,, s = ) + f, s ) s f, s ) f, ) f, s ) K Similarl, K K. W MinMa f U V where = ) +Θ ), 7) MSD) : Θ ),. r = Min f, ) ) f, ) + f, r ) r f, r ) f, ) + f, r ) K K K. ) For an given, the rogram MPS) MPD) are a air of second order smmetric dual nonlinear rogram involving cone treated in the roceeding section hence in view of assumtions ii)-v), heorem becomes alicable. herefore, for = we have ) s =, ) f, ) = 8) Θ ) =Θ ) 9) It remains to show that,, r = ) is otimal for MSD). If this is not the case, there eists V such that Θ ) <Θ ). But then, in view of the assumtions iv) v), we have Θ ) =Θ ) >Θ ) =Θ ), which contradicts the otimalit of,, s = ) for MSP). Hence, r=, ) is an otimal solution for MSD). Also, 5) 8) rove b), whereas φ, s, = ) = ψ, r, = ) follows form 6), 7) 9). As earlier, here to, the converse dualit theorem heorem 6) will be merel stated. heorem 6 Converse dualit). Let, r, ) otimal solution of MSD), also let be an i) f, ) be searable with resect to ii) f, ) be bonve in for ever, ), boncave in for ever, ), iii) f, ) be thrice differentiable in, iv) f is non singular, v) f, r ) ) is ositive definite. hen e) r = f) ) f = g) φ, s, = ) = ψ, r, = ) h), s, ) is an otimal solution of MSP). heorem 7 Self dualit). Let f : R n R m R be skew smmetric. hen MSP) is self dual. Further, if MSP) MSD) are dual rograms, s, ) is an otimal solution for MSP), then, s=, ), r=, ) are otimal solution for MSP) MSD) resectivel, φ, s, ) = = ψ, r, ). Proof. he roof follows along the lines of roof of heorem SPECIAL CASES n m n m If C = R + C = R + where R + R + are nonnegative orthants in R n R m, then the roblems SP)

7 Husain, Goel, Masoodi: Second Order Smmetric Mamin Smmetric Dualit with Cone Constraints IJOR Vol. 4, No. 4, ) 5 SD) will reduce to the following roblems treated b Mond 974): Primal P): Minimize G,, ) = f f + f, ) ) f + +,,, Dual D): Maimize H, q, ) = f f + f, ) ) q +, q ) f + f, q ),. It is to be remarked that can be deleted resectivel from the roblems P) D) as these constraints are not essential. If onl q are required the zero vectors, then our roblem SP) SD) become the following first order) smmetric dual rograms over cones studied b Bazaraa Goode 973): 5. Dantzig, G.B., Eisenberg, E., Cottle, R.W. 965). Smmetric dual nonlinear rograms. Pacific Journal of Mathematics, 5: Devi, G. 998). Smmetric dualit for nonlinear rogramming roblem involving η-conve functions. Euroean Journal of Oerational Research, 4: Dorn, W.S. 96). A smmetric dual theorem for quadratic rograms. Journal of Oerational Societ of Jaan, : Gulati,.R. Ahmad, I. 997). Second order smmetric dualit for nonlinear mied integer rograms. Euroean Journal of Oerational Research, : Kumar, V., Husain, I., Chra, S. 995). Smmetric dualit for minima mied integer rogramming. Euroean Journal of Oerational Research, 8: Mond, B. 974). Second order dualit for nonlinear rograms. Osearch, : Mond, B. Weir,. 989). Generalized conveit dualit. In: S. Schaible W.. Ziemba Eds.), Generalized Concavit in Otimization Economics, Academic Press, New York.. Na, S. 988). Inve generalization of some dualit results. Osearch, 5): 5-. Primal P ): Minimize f f f, ) C, C C, Dual D ): Maimize f f f, ) C, C C. Finall, if U V are emt sets = s r = q, hen MSP) MSD) will become, the roblems SP) SD) considered in Section 3. REFERENCES. Balas, E. 969). Minima dualit for linear nonlinear mied integer rogramming. In: J. Abadie Ed.), Integer Nonlinear Programming, North-Holl, Amsterdam.. Bazaraa, M.S. Goode, J.J. 973). On smmetric dualit in nonlinear rogramming. Oerations Research, ): Bector, C.R. Chra, S. 986). Second order smmetric self dual rograms. Osearch, 3: Chra, S. Husain, I. 98). Smmetric dual non differentiable rograms. Bulletin of the Australian Mathematical Societ, 4: